Discrete ABP estimate and rates of convergence for linear elliptic PDEs in non-divergence form.
Dienstag, 20.1.15, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We design a two-scale finite element method (FEM) for linear elliptic\nPDEs in non-divergence form. Besides the meshsize, a second larger scale\nis dictated by an integro-differential approximation of the PDE. We show\nthat the FEM satisfies the discrete maximum principle (DMP) provided\nthat the mesh is weakly acute. Combining the DMP and weak operator\nconsistency of the FEM, we establish convergence of the numerical\nsolution to the viscosity solution of the PDE.\n\nWe develop a discrete Alexandroff-Bakelman-Pucci (ABP) estimate which is\nsuitable for finite element analysis. Its proof relies on a geometric\ninterpretation of the Alexandroff estimate and control of the measure of\nthe sub-differential of piecewise linear functions in terms of jumps,\nand thus of the discrete PDE. The discrete ABP estimate leads to optimal\nrates of convergence for our finite element method under natural\nregularity assumptions on the solution and coefficient matrix.
Rate-independent damage models with spatial BV-regularization -- Existence & fine properties of solutions
Dienstag, 10.2.15, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
In this talk we address the existence of energetic solutions for a model of\npartial damage with a BV-gradient regularization in the damage variable.\nFurthermore, we discuss properties of energetic solutions that can be\nobtained in a setting where the damage variable is a characteristic function of \nsets with finite perimeter.